A GENERAL FIRST MAIN THEOREM OF VALUE DISTRIBUTION. I

A GENERAL FIRST MAIN THEOREM OF VALUE DISTRIBUTION. I

BY

W I L H E L M STOLL

University of Notre Dame, Notre Dame, Indiana, U.S,A. (1) Dedicated to Marilyn Stoll

The theory of value distribution of functions of one complex variable is well developed a n d has yielded beautiful results a n d applications. An i m p o r t a n t aspect is t h a t a non- constant meromorphic function can be considered as an open m a p into the l l i e m a n n sphere P, so providing a ramified covering of an open subset of P.

Compared to the theory for one variable, the theory of value distribution for functions of several complex variables is still in its infant stage. The dual aspect seems to be lost.

One w a y is to s t u d y the value distribution of a holomorphic function. Here H. Kneser [7] (1936) a n d [8] (1938) initiated a theory of value distribution for a meromorphie function of finite order in C =. This theory was later completed in [19] (1953) and gave an analogon to the theory of functions of finite order of one variable. The Kneser integral(2) substituted for the Weierstrass-product. I n [20] (1953), these results were applied to construct Jaco- bian and abelian functions similar to the construction of the a, ~ and ~o function in one variable. Also the Kneser integral was extended to functions of infinite order in [19]

(1953) a n d was applied in [14] (1964) to the theory of normal families of divisors in C ~.

Ahlfors, H. Weyl and J. Weyl developed a theory of value distribution of a holo- morphic m a p / : M - + P ⁿ of a R i e m a n n surface M into the n-dimensional, complex projec- tive space P=. T h e y were able to prove both Main Theorems a n d the Defect Relation.

These results can be found in [29] (1943). I n [21] (1953) and [22] (1954), this theory was united with the theory of Kneser to a value distribution of meromorphic m a p s / : M - + P ⁿ where M is a pure m-dimensional complex manifold. Both Main Theorems a n d the Defect

(1) This research was partially supported b y t h e National Science F o u n d a t i o n under grant N S F GP-3988.

(2) A different integral representation was used b y Lelong [9], [10] a n d [11], which enables him to give a new proof of the Cousin I I Theorem for positive divisors of fillite order.

112 WILHELM STOLL

Relation could be proved. I n [25] (1964) and [26] (1964), the First Main Theorem was extended to the case, where M is a pure m-dimensional analytic set in a complex manifold.

An application showed, t h a t an analytic set M of pure dimension m in C = is algebraic if and only if the 2m-dimensional Hausdorff measure of M N {z[ I~1 < r } is O(r~m) for r-~oo.

The other aspect of value distribution in one variable leads to the s t u d y of open, holomorphic maps ] : M ~ C m where M is a pure m-dimensional complex manifold. The fibers are discrete. F a t o u [4] (1922) and Bieberbach [1] (1933) found a biholomorphic map of C ~ onto an open, non-dense subset of C ~, whose Jacobian was constant. This discouraged the " m a p " approach. However, Schwartz [15] (1954) and [16] (1954) used topological methods to s t u d y this case and generalized the Ahlfors theory of covering spaces. Levine [12] (1960) found an unintegrated First Main Theorem.

Chern [3] (1960) integrated this theorem and obtained a result (Chern-Picard Theorem) which assures t h a t an open holomorphic map ]:Cm-+P m assumes almost every point in pm if the characteristic of ] grows stronger t h a n the new "deficit" term which appeared in his First Main Theorem. Hence, the difficulties which the Bieherbach map posed were overcome.

l~ecently, B o t t and Chern [2] (1965) have extended these methods to the s t u d y of sections in fiber bundles and obtained deep results.

Comparing the formulas in Levine [12] and Chern [3] with the formulas in [21], [22]

and Kneser [8], a great similarity becomes apparent. Therefore, the question of an united theory arises. Such an unification will be provided here.

L e t M and P be connected complex manifolds with dim M ~ m and d i m P = n . Let be a family of pure p-dimensional analytic subsets of P. L e t f : M ~ P he an holomorphic map. Suppose t h a t ] is general of order r = n - p in respect to 7, t h a t means, t h a t ]-1(S) is e m p t y or analytic of pure dimension q = m - r for every S E 7. The theory of value distri- bution is concerned with the size of ]-1(S) for SE 7. A typical statement would be: If ]

"grows" strongly enough, f(M) intersects " m o s t " elements of 7. Usually, the "nmst"

which results from a "_First M a i n Theorem" would be in the Lcbesgue measure sense, where upon the "most" which results from a "'Second M a i n Theorem" would mean "all up to finitely m a n y " . The theory of Kneser and Stoll belong to the codimension r = 1, the theory of Chern and Levine belong to the codimension r = n . Here, both theories will be special case of an unified t h e o r y for codimension r where 1 ~<r ~<Min (m, n), at least so far the "_First M a i n Theorem" is concerned. P will be the n-dimensional complex projective space, and ~ will be the Grassmann manifold of all p-dimensional complex planes in P.

L e t F be an (n+l)-r]irnensional complex vector space with a given Hermitian pro- duct ([). On every exterior product V[p] = V A... A F (p-times) an associated Hermitian

A G E N E R A L F I R S T M A I N T H E O R E M O F V A L U E D I S T R I B U T I 0 1 ~ . I . 113 product is induced if 0 < p ~ < n + l . I t defines a norm on V[p]. For a e V - { 0 } , define

~(a) = {zalz q C}. Then

r ( v ) = {e(a) I a e v - {0})

is the complex projective space of V and ~: V-{0}-~P(V) is the natural projection. De- note the natural projection of

V[p]-

{0} onto P(V[p]) also b y ~. Define

~'~+.t)(v)

= {CoA ... A c~[c.e v}_~ vho + 1].

Then (~(V) = e((~+x(V) - {0))

is the Grassmann manifold of the p-dimensional complex planes in P(V). Of course (~0(V) =

P(V).

If aE(~(V), then E(at) and E(at) are well defined by

E(a) = {3]3A a=0}, where Q ~ Q - - l ( a )

Z(at) = e ( ~ ( a t ) - {0}).

If ate(~(V) and

fle~q(V),

then ]]at:fl]] are well defined by

II~:Zll laA~l

= [ a ] ] b [ ' where aee-~(at), bee-l(fl).

Moreover, 0 < liar:fill ~< 1.

On a n y complex manifold, let d = ~ +~ be the exterior derivative with ~ as the com- plex component and ~ as the conjugate complex component. Define d ' = i ( ~ - ~ ) . On

V - {0}, define the non-negative

"projective"

form eo by

w($) = ld'dlog

[$].

Then one and only one positive

"projective"

form ~ of bidegree (1, 1) exists on

P(V)

such t h a t Q*(~)=w. Here ~ is the exterior form of a Kaehler metric on

P(V).

Define the pro- jective forms of bidegree

(p, p)

by

o)~ =~.~ eo A ... A o); 1

Define the

"euclidean"

forms on V by

Let

O<p<n

defined by

w~=-~ ~ A . . . A ~ 1 (p-times).

p.,

i ~(313); vr =~.t v A A v (p-times)

V ~ . . .

and a E ~ ' ( V ) . On V-E(at), a form d)(at) of bidegree (1, 1) is well-

alP(or) (3) = ld•

log [ a A 3[,

8 -- 662905 Acta mathematica. 118. I m p r i m 6 lo 12 a v r i l 1967.

114 w[ L~m~M STOLL

where aE~-I(a). One a n d only one non-negative form ~P(~) of bidegree (1, 1) exists on P(V) - ~(~) such t h a t

e*((I)(~)) = ~(~).

Define r = n - p . On P ( V ) - E ( ~ ) , define the Levine form of order r b y

1 r-1

Y r A/5

'-l-~'.

A(a) ( r - l ) ! .-0

The form A(a) is non-negative a n d has bidegree ( r - 1, r - 1 ) . F o r r = 1 is A ( a ) = 1. Define W ( s ) = ~ i f s E N .

L e t M be a complex manifold of pure dimension m. L e t g be a differential form of bidegree (q, q) a n d class C 1 on M with d Z = 0. Suppose t h a t

0 < m - q ffi n - p = r < M i n (m, n).

L e t / : M ~ N be a holomorphic map, which is general of order r (in respect to (~(V)).

F o r every z E / - I ( $ ( a ) ) , an intersection multiplicity vf(z; ~) is defined, which is a positive integer. The function v1(z; :r of z is constant on every connectivity component of the set of simple points o f / - l ( / ~ ( a ) ) . As a consequence of a residue formula (Theorem 4.4) the following result is obtained.

T ~ E UNINTEGRATED FIRST M A I N T H E O R E M ( I ) . Let H be an open, relative com.

pact subset o / M whose boundary S is either empty or a smooth, (2m-1)-dimensional sub- mani/old o / M which is oriented to the exterior o / H . Take o~ E (~'(V). Suppose that

is integrable over S. Then

$(ar = d ' l o g

IIt: ,11

A I*(A(a)) A z

1 1

~(,,)+ [ , , . ( , ; , , ) z = V ~ t*(&)AZ.

27~ W ( r - l) as jnnt-R~(,,))

F o r r = m (i.e., q = 0 ) a n d Z = I , this is the First Main Theorem of Levine [12]. I f M is compact and H = M, then S = O a n d

- ' & , , , ~,,-(z; =)

Z = W-~ l*(&.) A Z

is constant on (~(V). I f r = n (i.e., p = 0 ) , then ( ~ ( V ) = P ( V ) a n d / : M - + P ( V ) is a n open (1) See Theorem 4.5.

A GENERAL FIRST MAIN THEOREM OF VALUE DISTRIBUTION. I 1 1 5

holomorphic map of pure fiber dimension q(1). In ZI,(2) the continuity of the fiber integral

hi(G; o~) = fi_l(~(~))

vr(z; ~) Z

was proved. I-Iere, it turns out that

ni(G; o~)

is constant, if dg =0 and if the image mani- fold is P(V).

Now, suppose that Z is non-negative on M. Let G and g be non-empty, open, relative compact subsets of M with smooth C~176 F of G and ~ of g, both of which are oriented to the exterior. Assume that ~ c G. Let yJ be any continuous, non-negative func- tion on G, such that yJ[F = 0 and Y~lg = R > 0 are constant, such that ~ is of class Coo on

G - g and such that 0~<yj(z)~<R for zEG. Define the

characteristic

of / by 1 /" *(oJr )A x.

For ~ E (~(V), define the

compensation/unctions

by

m1(F; ~)

2~ W(r -

I 1 1) f r log ~ /*(A(a)) A d'y~ A Z,

mI@; ~) - 2 ~

W ( r -

1) log I*(A(~)) A d ' ~ A Z.

Define the

valence/unction

by(3)

vs(z:~)

The integrands of these integrals are non-negative. This is not true for the

deficit

1 1 fv. 1

AI(G; a) 2~

W ( r -

1) __l~ ~ / * ( A ( a ) ) A

dd'~v A X.

These assumptions imply the

First Main Theorem(4)

Nr(a; ~r a ) - m1(~,; a) =

Tr(G )

+At(G; a).

The theory of [21] and [22] is obtained by choosing

r = l ( q = m - 1 )

and d d ' ~ A g = O

(1) S u c h a m a p is also called q-fibering.

(2) Stoll [27] is d e n o t e d b y I a n d Stoll [28] is d e n o t e d b y I I , b e c a u s e t h e s e p a p e r s a r e n e c e s s a r y p r e p a r a t i o n s for t h i s p a p e r a n d wore w r i t t e n s i m u l t a n e o u s l y .

(s) I f q = 0, t h e i n t e g r a l is a finite s u m . (4) See T h e o r e m 4.8.

116 w R ~ . ~ STOLL

o n G - y . T h e t h e o r y of H . W e y l a n d J . W e y l [29] is o b t a i n e d b y choosing m = r = 1 a n d Z = I w i t h d d ~ p = 0 o n G - ~ . T h e t h e o r y of H . K n e s e r [8] is o b t a i n e d b y choosing r =

l = n ( q = m - l , p = O ) a n d M = C m w i t h Z=Vm_I, w h e r e G = { 3 1 131 < r ) a n d g = { ~ ] 131 < t o ) w i t h O < r o < r a n d

1(i, 1)

~ 0 ( 3 ) - 2 m _ 2 312m_ 1 r2~_ ~ on G - ~ .

T h e

theory

of Chern

[3]

is o b t a i n e d b y choosing r = m = n ( q = O = p ) a n d M = C m w i t h g = l . Moreover, O, g a n d ~0 a r e c h o s e n a s in t h e t h e o r y of H . K n e s e r . H o w e v e r T h e o r e m 5.12 shows t h a t t h e C h e r n - P i c a r d T h e o r e m c a n b e o b t a i n e d for o t h e r choices of ~.

I f / : M ~ P ( V ) is a n o p e n m a p , t h e n / is g e n e r a l of o r d e r s for e v e r y s in 0 < s ~ n . I n a d d i t i o n s u p p o s e t h a t M is a c o n n e c t e d , n o n - c o m p a c t , p s e u d o c o n v e x m a n i f o l d , w h e r e p s e u d o c o n v e x m e a n s , t h a t a n o n - n e g a t i v e f u n c t i o n h of class C ~ e x i s t s o n M s u c h t h a t dXdh>~O a n d such t h a t

G(r) = {z [h(z) <r}

is a r e l a t i v e c o m p a c t s u b s e t of M if r >0.(1) Choose a n y s u c h a p l u r i s u b h a r m o n i e f u n c t i o n h.

Define

Z8 = 1 d• A ... A d• (s-times).

8 [

T a k e r o > 0 such t h a t g = G(ro) 4 0 a n d such t h a t dh # 0 on G(ro) -G(ro) = 7 - F o r r > to, d e f i n e

I

^O

V,(z) = r - h(z)

[ 7" - - r 0

if z E M - G(r), if z E G(r) - g, if zEg.

T h e c h a r a c t e r i s t i c of [ for t h e o r d e r s is 1

Ts'r(r) = ~ ) ) f G(r) Y~r/*(ws) A Zm-~

i f r >/r 0. T h e f u n c t i o n T8.I is d i f f e r e n t i a b l e for r >~ r 0 a n d i t s d e r i v a t i v e is

D e f i n e An-l,f(r)

(~I = l i m s u p - - , r--~ T~,r(r)

1 f di~.

b ( M ) = W ( n ) (M)

(1) M is a Stein manifold if and only if such a function h with the stronger condition d~dh > 0 exists. See Grauert [5], [6] and Narasimhan [13].

A G E N E R A L F I R S T M A I N T H E O R E M O F V A L U E D I S T R I B U T I O N . I 117 Then

b(M)

is the normalized measure of the image of / in P(V). Obviously

O<b(M)<~1.

Then the following generalization of the Chern-Picard Theorem [3] holds:Q) THEOREM.

Suppose that Tn.1(r)-->~ ]or r ~ . Then

- n + 1 ~ 1 O<'(1-b(M))<~m 4~ ~=ov-+l ~f"

Especially, i/~f= O, then ] assumes almost every "value" o/P(V).

As a preparation for the proof of the First Main Theorem, the limit of certain integrals is obtained in section 1. The intersection n u m b e r vr(z; a) is introduced a n d studied in section 2. I t s definition is reduced to the multiplicity of a certain open holomorphic m a p gl. I n section 3, the Levine form is studied. I n section 4, two integral theorems are p r o v e d which yield the First Main Theorem in its unintegrated a n d its integrated form. I n section 5, the

"spherical"

representation (Theorem 5.8) of the characteristic a n d of the deficit is proved for open maps. The Chern-Picard Theorem [3] is extended to open maps.

The results of B o t t a n d Chern [2] have not been considered in this paper, because almost all of this research h a d been done as [2] became available to me.

w 1. The existence and the limit of certain integrals

The proof of the First Main Theorem requires some highly technical results a b o u t the existence a n d the limit of certain integrals. The proof of these results shall be given here to avoid an undue interruption of the later representation.

The concepts a n d notations of I a n d

11

shall be used. F o r convenience sake, some are recalled here.

1. L e t V be a complex vector space of dimension m. Define Vm= V • ... • Y (m-times).

Then the general linear groups

GL(V)

= {(cl ... Cz)lq A ... Acm*O}

which is the set of all bases of V, is open and dense in V z.

2. A function (I): V x V~t3 is a

Hermitian product

on V, if a n d only if it is linear in the first variable, if (El ~)) = (t) I ~) for all ~ E V a n d t) E V a n d if (~ [ ~) > 0 if 0 # ~ E V. De- note I~[ = ~ / ( ~ ) . Then V becomes a complex Hilbert space. An element (Cl ... cz) E

V z

is said to be a n orthonormal base if and only if (c,[ c,) = 0 for all/~ =i:v as [c,[ = 1 for all/x.

The set II(V) of all orthonormal bases of V is a non-empty, connected, real analytic sub- manifold of GL(V).

(1) See Theorem 5.13.

118 WILHELM STOLL

3. If not otherwise directed, the space C~= C • • C is thought to be endowed with the Hermitian product defined b y

(~1 ~) = ~ x~,~, if ~ = (~1, ..., ~m), ~ = (Y~ . . . Y,n)-

Then the base (r ... era) is orthonormal, where ep=(~l~ ... Om~) and ~ = 0 if p~=r and

(~/q, =

1.

4. The group ~(m) of all permutations ~ of {1 ... m} operates effectively on V m, G L ( V ) and U(V) b y setting

Y'~(C 1 . . . era) = (C~(1), ..., C~(m)).

L e t ~(q, m) be the set of all injective and increasing maps

~ : { 1 .... , q}-~ {1 ... m}

T h e n ~E~(q, m) defines one and only one permutation ~ E ~ ( m ) such t h a t _~(~)=~(~) for

~ = 1 ... q and ~(r) < ~ ( ~ + 1 ) for ~ = q + l ... m - 1 .

5. Let M be a pure m-dimensional complex manifold. The set ~M of all biholomorphie m a p , : U~-~ U~' of an open subset U~ of M onto an open subset U~ of C m is the complex structure of M. Then ~ = (z~, z~), where z ~ - , are holomorphic functions on U~. Let d be the exterior derivative on M. For ~ E ~(q, m) and y~ E ~(q, m) define

~ = ~ = dz~(;) A ... A dz~(q), / \ i q

~/~=l'/~m= ~ )

dzq~(l) A dZv(l) A ... A dzq~(a) A d~,~(q) on U~. Then ~ = ~ = ~(1) A . . . A d~v(q),

6. If X is a differential form of bidegree (q, q) on a subset A of M, if ~ E ~M with A N U~ :~O, then

rCe~.(q.m) ~e~(q.m)

on A N U~, where Z ~ are uniquely defined functions on A N U~. Z is said to be real, if Z=2~; this is the case, if and only if Z ~ = Z ~ for all % ~0, ~. If Z has bidegree (m, m) on A, then {t}=~(m, m), where t is the identity and Z=Z,~/,. The set of all real forms of bidegree (m, m) on A is ordered at a E A such t h a t Z ~ at a if and only if Z,~(a)~<~(a) when ever ~ E ~M with a E U~.

A G E N E R A L F I R S T MAI-~T T H E O R E M OF V A L U E D I S T R I B U T I O N . I 119 7. L e t M and N be complex manifolds. L e t ]:N-->M be holomorphic. Then, every form Z on M induces a form/*(Z) on N. The map a* is a homomorphism, which preserves degrees, bidegrees and conjugation and commutes with d, 0, 0 and d • where 0 is the com- plex component and 0 is the conjugate complex component of d, and where

d 9 = i ( ~ - ~ ) .

8. Let M be a pure m-dimensional complex manifold. Let 0 < q < m . Let Z and ~ be real differential forms of bidegree (q, q) on MI Then g~<~ at aEM, if and only if for every smooth, pure q-dimensional complex submanifold N of M with inclusion ~:N-->M, the inequality J~(Z)~<J~(~) holds at a. (Observe, t h a t 6. applies, because ?*(Z) and ?'*(~) are again real and of bidegree (q, q).) A partial order on the set of all real differential forms of bidegree (q, q) on M is defined, especially, the concept of positive, negative, non-negative and non-positive forms at a are defined. If A c _ M then g~<~ (resp. Z<~) on A if g~<~

(resp. g < ~ ) at every a E A . Hence g is positive (negative, non-negative or non-positive on A) if this is true at every point of A. The forms ~ are non-negative on Ua.

9. L e t M be a pure m-dimensional complex manifold. L e t y) and g be forms of bide- t r e e (q, q) on M with O<q<.m. Let Z be non-negative on M. Then I~vl ~<g means, t h a t for every smooth, pure q-dimensional, complex submanifold N of M with inclusion ~N:N-+ M, the inequality

l <J (z)holds.

10. L e t V be a complex vector space of dimension m with a Hermitian product (I).

l~or $ E V, define

v = ~ (d~ [d3) i

vp = ~. v A ... A v (p-times). 1

Then v is the associated form of the K/ihler metric defined b y (I). Moreover, v~ is positive if 0<p~<m. The forms v and v~ are called the euclidean/orms to ([). If c--(el ... Cm) is a base of V, define ~: V ~ C m b y a(c~)=e~ for # = 1 .... , m.(~) Then g~:Cm-~C is defined b y g~(z~ ... zm) =z~,. Then z ~ = : ~ o ~ : V->C are the coordinates associated to c. I t is

~= ~z~($)Cu for ~ V .

For ~0~Z(lo, m) and ~)~Z(I~, m), define r and ~ = ~ / ~ v , if Clear, write ~ = ~ e ~ 9 and

~/~ = ~ and call these the forms associated to c. Define g ~ = (ct~l c~) then (1) Here e/~ = ((~/n ... (~/~m) where (~/~v = 0 if/~ ~ and where (~/~ = 1.

120 w H.nELM 8TOLL

m

v = - ~ g , , d z , A d~,.

2 l~.pffil

I f a n d only if c E 1I(V) is o r t h o n o r m a l , t h e n g#v = ~#v a n d

rn

11. L e t V be a c o m p l e x v e c t o r space of dimension n + 1 w i t h a H e r m i t i a n p r o d u c t (I). L e t P(V) be t h e associated p r o j e c t i v e space a n d let O: V - { O } - + P ( V ) be t h e residual m a p . T h e n O-I(o(Q)) = {zal0 :#2: E C}, w h e r e a 4=0. On V - {0}, define

| 1 i

= z ~ dd• log I$[r:i = ~ OR log (3 [3).

(,O

T h e n d o ) = 0 a n d O J l ~ - - i (3[3) (d$1d3) - (d315) A ($1d$)

2 I 1'

This f o r m is n o n - n e g a t i v e a n d called t h e projective/orm to (I) on V. Define e% = ~ co A ... A co (p-times) 1

p :

T h e n o)~ is n o n - n e g a t i v e a n d called t h e p r o j e c t i v e f o r m of bidegree (p, p) on V. On P(V), one a n d only one exterior f o r m ~ of bidegree (1, 1) exists such t h a t Q*(oJ) =co on V - (0}"

T h e f o r m ~ is real analytic, positive definit a n d d ~ = 0. I t is t h e exterior f o r m associated to a K a e h l e r m e t r i c on P(V). Define

1 ..

~p = ~ co A ... A ~ pl (p-times)

Then on V - {0}.

12. L e t M b e a n oriented, real m a n i f o l d of class C ~176 L e t N be a n oriented, real sub- manifold of p u r e dimension n a n d class C k w i t h k >/1. L e t ] N : N - ~ M be t h e inclusion m a p . L e t yJ be a f o r m of degree n on a subset A of M such t h a t ?'*(~0) is integrable o v e r A N N .

Define

13. L e t M be a c o m p l e x manifold. L e t zV be a p u r e n-dimensional a n a l y t i c subset of

A G E N E R A L F I R S T M A I N T H E O R E M O F V A L U E D I S T R I B U T I O N . I 1 2 1

M. Let N be the set of simple points of M. Let yJ be a form of bidegree (n, n) on a subset A of M, which is integrable over A N N . Define

If A is compact and if ~o is continuous on A, these integrals exist.

I n the following lemmata, the spaces C m and C p with O<p<<.m appear often. The forms vq, ~%, ~ and $~ are formed for the natural coordinates. Those on (~m shall be dis- tinguished from those on C v b y a lower bar: %, wq, V~ and $~.

L~,MMA 1.1. Let M be an open, non-empty subset o / C m. L e t / : M ~ C v be a holomorphic, q-fibering map.(1) Let s E N. Suppose that 1 <<. s < p = m - q <<. m. Let ~ be a non.negative integer.

Let Z be a dif/erential form of bidegree ( m - s , m - s ) on a compact subset K of M. Suppose that the coefficients of Z are measurable and bounded on K. Take ~E~(s, p). ~ o r ~ER with 0 < ~ < 1, define

L(e ) = {z [89 ~< ]/(z)] ~< e with z e g ) ,

1 ~ 1 ,

I (~)= fr~(o)(log []]) V]~/~] f ( ~ ) A Z,.

Then I(~)-->0 /or ~-+0.

Proof. Without loss of generality, ~ ( v ) = v for v = l , 2 ... s can be assumed. Now, f = (fl .... , fv), where fg is holomorphic on M. Then

f*(~q~) = (89 ~ df~ A d / A . . . A df~ A d/,.

An open neighborhood H of K exists, such t h a t H is compact and contained in M and such t h a t H is a finite union of balls. On M - K , define Z b y setting g(z)=0. Then

~e T;(m--s,m) ~eyXm--s.m)

A constant B > 0 exists such t h a t ]Z~p($) ] ~< B if $ fi M for all ~ and fl in ~ ( m - s, m). Define F~ =~(/1 . . . /~) for ~ e ~ ( m - s , m ) .

~(Zct(rn - s + l ) , - . . , Z~(m) )

Then [ f * ( ~ ) A Z] = ] ~. ~ sign a sign flF~_F~ X~lvm tte~(m--s.m) fl~.~.(rn--s,m)

_<B

t~E~(m--s,m) .Sea(m--s, m)

Q) A holomorphic map ] : M ~ N is said to be q-fibering if and only if I-l(I(a)) is a pure q-dimen- sional analytic set for every a E M.

122 ^{W I I} AqI"II~,T,M S T O L L

\ 8 ] ~E~:(m-s,m)

(9

= B d l i A d f i A A d ] , A d / , A v _ m _ .

8 2 " ' "

F o r 0 E R with 0 < ~ < 1 define

I/I! It I'"/*(n,.) ^

~_,,-~

Then L(~)_~ T(0 ) a n d

I(O)<.(?)B.J(~),

if 0 < 0 < 1 . Therefore, it suffices to prove

J(~)->O for ~-*0.

Define g:~2'->ff b y ~(z~ ... %)=(z~ ... z,). On C ~ distinguish the standard forms b y a dash: v~, o~, ~?~ a n d ~ . Then

gfzeo/:M.-,C"

is a m a p of pure fiber dimension

q + p - s =

, t , 9

m - s , hence ( m - s ) - f i b e r i n g . Moreover, ~ r (v,) a n d / ( ~ ) = g * ( v , ) . According to I I Proposition 2.9 is

1 ~ 1

, , - , ,

[~) I/T~ ~'-')"

~ea-,(ro) n T(e), then Irol = la(~)l <

I/(~)I <e-

~e-oe

(*)

4" 1 "

<e-~fl~.,~.(f._.,~,o~,o,".(I~ ~-,..)":

for 0 < ~ < 1 .

Because g-l(O)_/-1(0) with dim g-l(O) -- m -- s a n d dim/-1(0) ---- m - p and m - p < m - s,

A GENERAL FIRST MAIN THEOREM OF VALUE DISTRIBUTION. I 123 / d o e s not vanish identically on a n y branch of g-~(lv) if iv E C'. A constant 2' > 1 exists such t h a t [/($)1 ~<F if SEH. According to Theorem 4.9, the integral

G(w)= f _a(~)ng (l~ ~/])"%v-~-s

is a continuous function on ~s. Therefore, a constant U > 0 exists such t h a t 0~<G(lV)<d if Iml < 1 L~t 0~<~<1 and lvEG ~ with Jml <1. Then

1 ~

(

log--j/i)Vm_s~G,~iv,<C. F

N O W ,

for 9-~0 shall be proved.

Take e > 0. Define

U

~eg~f-X(O)

Because H is compact and contained in M, an open neighborhood H o of H exists such t h a t

~ 0 is compact and contained in M. Then e 0 > 0 exists such t h a t

A(~i~_A(~o)~_Ho

if O<~<e,.

Therefore A(~) is compact if O < t ~ e o. Take SE.~(89 A sequence {~v},EN Of points in A(89 converges to 3. Hence t~vEHA/-I(0) exists such t h a t [3,-t~,[ <89 Because H is compact, a convergent subsequence { t~v~}~eN exists, where v~-~ c~ for ~t-* c~ and where t)=lima_~D~aEHN/-l(0). Then J$-DI < 89 Hence 3EA(e). Therefore .~( 89 Take a continuous function 28 on G m with 0~<)t~<~l such t h a t )t~ has compact support in A(e) and such t h a t 28($)= 1 if $

EA(89

Suppose, t h a t a sequence {~v}vGN Of real numbers with 0 < ~ < 1 converges to zero such t h a t T(~p)-A( 89 for each v E N. Pick 3v E T(~)-A( 89 for each ~ E N. Then 3~ E H.

A subsequence {3va}~, with ~-~ co for 2-~ c~, converges to a point t)EH-A( 89 More- over, DC(3va)J ~<Q~ for 2EN. Hence /(t~)=0. Therefore, ~)EH

~/-~(O)-A(89

^Contra-

diction! Hence, for each ~ in 0 < e ~ 0 ; a number ~0(e) with 0 <~0(e)< 1 exists such t h a t T(~)_~A(89 if 0 <~ ~<~0(e). Consequently,

ff ]iv] ~<1 and O<O<Qo(e ).

124 w-m ~r~M STOLL

Because ~ is continuous, because H is the union of balls and because / is not iden:

tically zero on any branch of g-l(0), II Theorem 4.9 implies

@--~0 J g-- l(@~)n T(q)

If O<e'<e, then

A(e')~_A(e).

Moreover, n0<,<~~ A(e)=/t N/-1(0), which is a subset of measure zero on g-l(O). Hence

for e--> 0. Hence

fo- l(0)n~n A(8) ~: (l~ ~ ) ~vm-s-~0 l im0 fa-l,q~)n,,,) % (l~ ~]/) -v~-' = 0"

1

Because the integral is non-negative, limQ_~o can be replaced by limq_,o. Because the integral is uniformly bounded, the theorem of bounded convergence implies in (*) that J(~)-*0 for ~-*0, q.e.d.

L E M M A

1.2.

Let M 4 0 be an open subset o/C m. Let/:M-->C p be a holomorphic, q-fibering map with q = m - p and 0 <p <~m. Let K be a compact subset o / M . Let Z be a differential /orm o/bidegree

(q,

q) on K with bounded and measurable coe//icients on K. For

0<Q <1,

define

L ( ~ ) = { $ I s E K ,

2 "~Q--<[/($)I ~<0},

f 1

z(e)

⁼

[z A l*(v,) 1.

L (q)

Then a constant

D > 0

exists such that

[I(0)[ ~<D i / 0 < 0 < 1 .

Proo/.

At first, consider the ease q>0. Let ~0E~(p, p) be the identity. Continue Z onto M by setting

Z(z)=O if z E M - K .

Denote/=(/1

...

fr). Construct H, B, T(~),

J(~),

~ = v ~ , F > l , g=Id:~2'->C r and g = / as in the proof of Lemma 1.1, where

s = p

and

=0. Then

I ( ~ ) ~ ( ; ) B J ( ~ )

if O<Q<I

A G E N E R A L F I R S T M A I N T H E O R E M O F V A L U E D I S T R I B U T I O N . I 125

and J(~o) = fr(e) [~]~p [* (v) A vq

= fol2~l~l<e ] l]'o ( ffl(ro)n~'fv-q) v2'

\jr-](ew)n~ - /

Because I~ is the union of finitely many balls, Sf-l(ro)n~f__.q is eOHtinuous on t~, hence bounded by a constant Do > 0 on the unit baH. Hence

fr ~rv-q Do

0 ~ -~(oro)n~

if [tO[ ~ 1 and 0 < 0 < 1. Therefore

O<~J(q) <~4~.~. Do=D x

and

I(q)<~(p)BD I = D

f f 0 < q < l .

As the second case, consider q =0. Then p = m. The map / is light. According to I Lemma 2.5 a constant D o > 0 exists such t h a t

ns(K;

m) = ~ ~r(3; to) ~< D o if to e C ~.

~ e E

Because q =0, the form g is a function, which is bounded and measurable on K. Hence B > 0 exists such that [g($)[ < B if 3 e K. Therefore

5 vt(3; m)]Z(3)[

<~PoB.

~r

According to I I Lemma 2.8 is

f z 1 fe (: ,A3;ro)Jz(3)l)

1

4 m /" :r/: m

<~':ygmDo B l Vm = 4reDo B =D if O < e < l ,

q.e.d.

LEMM.A 1.3.

Let M g=~ be open in C m. L e t / : M ~ be a holomorlghic, q./ibering map with q =m-19. Let ~, s and t non-negative integers such that

l <~s<<.19<-~m, l <~t~p<~m, s+t<219.

126 WU~H~T~M 8TOLL

Let q) be in ~E(8, p) and let ~ be in ~(t, p). Let K be a com1~ct subset o/ M. Let Z be a di//eren.

tial form on K with bounded and mea~rable coe//icients on K. Define a = m - s and T = m - t , Suppose that Z has the bidegree (a, z). For 0 < Q < I , define

" <0},

Then l(e)-,0/or e~0.

Proo/. Without loss of generality, it can be assumed, that 1 ~<s < p and 1 ~<t ~<~o, Then, holomorphie function/~ exist such that / = (h ... /~). Then

]*(~) ~ dfr A . . . A d/~r and /*($~) = dry(1 , A ... A d]va,.

]3ounded and measurable function Z~$ exist on K such that

~eTs fle~(v,m)

A constant B~>0 exist~ such that IZ~p($)l < B for s e K and ae%(a, m) and fle~:(~, m).

Define

6',~ = a(l~,(,),::.,h,<,)) if flfi~:('t:, m).

~(z~(,+ 1) ... z~(,,,))

Then

II*r

~ ~ signocsign~F~O~Z.$lvm

~e?g(a.m) I~(~.m)

<B 5 Y IF~lla, lvm.

z" (e) = f, (0, l l . o 1 \~' ~1

Define

[og ) v.

/ 1"~ ~" 1

= Lo, t

Then O ~ I ( ~ ) ~ B ~ S f ( l o g l ~ " 1

~o~(o.m) ~,~(~.m)

.~,~), I11! I11 '+' I~1 I~l_~..

< B E ~ I , ( q ) t ' J ~ ( ~ ) 89

~ e%(a, rQ ~6e~(v,m)

A G E N E R A L F I R S T M A I N T H E O R E M O F V A L U E D I S T R I B U T I O N . I 127 According to Lemma 1.1, I~(~)~0 for ~ 0 . If t < p , then Jp(~)->O for ~ > 0 according to Lemma 1.1. If t = p , then J~(~) is bounded on 0 < ~ < 1 . Hence I(~)-*0 for 0 > 0 in both cases; q.e.d.

LEMMA 1.4. Under the same assumptions as in Lemma 1.3, the integral

exists.

Proo[. Without loss of generality, 0 < s < p can be assumed. Define L(~), I(Q), F~, Gp, I~(Q), Jp(Q) as in Lemma 1.3. Then Jp(~)-~0 for 0-+0 or JB(~) is bounded in 0 < ~ < 1 . I n either case, a constant D I > 0 exists such t h a t ]Jp(~)] ~<D 1 for 0 < ~ < 1 and flE~:(~, m).

Moreover,

F [ 1 \ s~ 1

1 F / 1 \~k+4 1

<~(logl/o)'JL(o(l~ ~ / * ( ~ v v ) A ~ a~'

where f L ( 1 \~,,+4 1

for ~-+0 according to Lemma 1.1. Hence a constant D 2 > 0 exists such t h a t I~(~)~< D s if 0 < 0 < 1 and ~E~(~, m). Hence I~(~)<(log 1/~)-4D2 for 0 < Q < I . Then

~e~s m) fle%(v.m)

1 B 1

for 0 < ~ < 1 , where B 1 is a positive constant. If 1 < n E N , then

~< B1 (~og 2)Sn s s ~ ,

where Bs is constant. Define K s = {$15 E K and

I/(3) 1<89

and KI = K - K s. Then K 2 = O~=~L(1/2 n) where L(1/2n+I) NL(1/2~)={$ I l/($)]=1/2 ~+1} f i E is a set of measure zero. Hence

= n = l =

128 WILHELM STOLL

Hence (log 1/111)" a/Ill '+'

1"(r ~)

^ x is integrable ^{o v e r} K~. Clearly, it is integrable over K x. Hence, it is integrable over K, q.e.d.

Now, the case s = 0 has to be treated.

L E y ~ 1.5. Let M + O be open in C m. Let [:M-~CP be a holomorphic, q-/ibering map.

Let O < p = m - q < ~ m . Let u be a non-negative integer. Let Z be bounded measurable ]arm o]

bidegree (m, m) on a compact subset K o / M . Far ~ e R with 0 < Q < I , define

~< 11(3)1 <o},

L ( 0 ) = { 3 1 ~ e K , ~

I(e)= yL,~)(l~ ~)'lZl"

Then I (~ ) ~ 0 / o r ~ ~ O. Moreover, the integral

exists.

Proo[. I f / ( ~ ) + 0 for all ~EK, the lemma is t r i b a l . Suppose t h a t A = K N]-l(O)~e~.

A constant B > 0 exists such t h a t ]ZI ~Bvm on K. Define E -- {w

I

I w I ~ 1 }. Define

~0:M • C-~C and g : M • C-~M as the natural projections. Then v~($, 0) = 1 for s e M . More- over, q0-x(0)=M • (0} and ~-~(0) N (K • E ) = K • {0}. Define j : M ~ M • C b y i(3)=($, 0).

Then g o ~ = I d : M - + M is the identity. Define g=]o~. Then go~=[. Moreover, g I = ~-x(0) fl g-l(0) N (K • E) = A • {0} =~O.

Apply I I L e m m a 4.8 using the table

There M m N p q ] ~ g b g K K

Here M• m + l C 1 m qJ ~t g 0 ~r*(v m) K • E K

y, ,,, Ilog Igl I" =* ('2_,~)

Hence J = - l ( o ) : f l ( K X E )

exists. Because j : M - ~ M • {0} is biholomorphic,

f K ~V

J =

]log I/ll

_~

exists. Because 9( is measurable and ]Z] <~Bvm. The functions (log 1/]]])"• are integrable over K.

[(log

X/lll)l"lx I and

A G E N E R A L F I R S T MA't"I~" T H E O R E M O F V A L U E D I S T R I B U T I O N . I

D e ~ e A(O) = { , I , e K , I1(,)1 < o } for 0 <O < 1. Define

129

Because Oo<q<1 A ( 0 ) = A is a set of measure zero, and because A(p')c_A(o) i f 0 < ~ ' < ~ < l , the integral S(~)-*0 for ~-~0. Because 0~<I(~)~<S(~), also 1(~)-~0 for ~-+0, q.e.d.

Of course, this lemma can also be proved direqtly.

L~.MI~IA 1.6. Let M . O be open in C m. L e t / : M ~ be a holomorphic, q./ibering map with q f m - p . Let ~ be a non.negative integer. Let tEN such that $ <~p <.m. Taker 2 in ~:($, m).

Define T = m - t . Let g be a diHerential /orm on the coml~d s~bsd K of M with bounded and measurable coeHicients on K. Suppose that g has bidegree (m, ~). F o r 0 < ~ < 1, define

x(e) = J ~(,),l~ ~ I*(r A z.

Then 1(0)~0 /or ~ 0 . Moreover,

exist.

1 " 1 , 1 " 1 ,

L(log ~) ~, (t,)Az a~d f.(log m) ~'! (r

Proo/. L e t ~ e ~(m, m) be the identity e(v)= r for u = 1 ... m. Bounded and measurable function Z# exist on K such t h a t

# ~ ~(~. m)

A constant B > 0 exists such t h a t I g#l ~< B on K. F[olomorphic flmotions /# exist such t h a t / = (/1 ... /~). Then

1"(~,)

=at,(,, ^ .... ^

ah,,.

Define Gp O([~a), " . , ]~(o) if fle ~(v, m).

8(z~(~+1) . . . z~(,o)

Then

It*(t,)Azl=l • sign#O~Z~lu,,<<.B E IOpl_,,.

# G ~(T, m) # G ~(T, m)

8I" - - 6 6 2 9 0 5 A a a mag~matica. 118. I m p r l m ~ le 12 a v r i l 1967.

130 W I L H E L M S T O L L

1 2 1 ,

Define J~(0) =

f.(0,~lG~l ~_~= fz,.)lT~[ (v.)^ ~.

I ' ( ~ ) = ,o, log]-]] Vm, f [ 1 "~2"+' I , ( 0 ) = J z,q,~l~ 1-/]) --Vm"

Then I, (0)-~ 0 and ~ (0)-~ 0 for 0-~ 0.

If t<io, then J p ( 0 ) ~ 0 for 0-~0; if t=p, then Jp(0) is bounded in 0 < 0 < 1 . In either case, a constant D I > 0 exists such that Jp(o)<D1 for 0 < 0 < 1 . A constant D z > 0 exists such that I,(0),.<D 2 if 0 < 0 < 1 . Then

1 ~ 1

O<"I'o)<"Be,z~(,.~)fr(o(l~

~IG'I v-

< B 5 LIepJ~(e)k

#~:(,.m)

Hence, I ( ~ ) ~ 0 for 0-~ ~ . Moreover,

1

1'(0) <

(log

1/0)' l'(q) ~< - - D~

(log 110) 4 for 0 < 0 < 1. Hence

VD~ (log i/0) ~ = 0og 1/0) * for 0 < 0 < 1, where B x is constant. If 1 < n E N, then

I ~< B 1 (1-~g 2) 2 n~ ^{_ B2} ~2

where B~ is constant. Define K , = { 3 1 3 e K and I]($)1 ~<89 and K I = K - K v Then K~=

[3 ,~176 L(112 n) where L(1/2 n) N L(1/2 m) is a set of measure zero if n 4=m. Hence

f x ( __1 ~'_1_1 ( 1 ) r162 1

logl/l/ ill, l/*($~)^xl= ~

^I

<B2n~__l--~< oo.

Therefore 0og

11/t])"ll//17*($~)^z

and its coniuga~ are integ~able over K,. ~early, they are integrable over K v Hence, they are integrable over K = K 1 0 K~, q.e.d.

P~OPOSITION 1.7. Let M be a complex mani/old o/pure dimension m. L e t / : M ~ f 2 be a holomorThlc, q-/ibering map with q = m - p , where O<p<<m. Let ~, s, t be non-negative integers such that

A G E N E R A L F I R S T MAIN T H E O R E M O F V A L U E D I S T R I B U T I O N , I 131 O<.s<~p, O<~t<~p, s + t < 2 p .

De/ine a = m - s and v = m - t . Let K be a compact subset o / M . Let % be a di//erential /orm on K whose coe//icients are measurable and locally bounded. Suppose that g has bidegree (a, T). Let q) be a di//erential ]orm o/bidegree (s, t) on ~ with measurable and locally bounded coe//icients. Suppose that/or every Q > 0 a measurable/unction h o is given on M . Suppose that these/unctions are uni/ormly bounded by a constant B on K, that is ] ho(z ) [ <~ B i / ~ > 0 and z E K. For ~ > O, de/ine

Ill]

III ~+~

Then I(e)-~O /or Q-~O and

2= loglll! ii[.+,t*(~)Az

is integrable over K.

Proo/. The form Z is integrable over K, if a n d only if every point z 0 E K has a c o m p a c t neighborhood U such t h a t Z is integrable over K (I U. Therefore, it can be assumed t h a t M # O is open in C m.

1. Case: s = t = 0 . Then ~0 is a function a n d / * ( ~ ) = ~ o 1 is a function on M, which is bounded a n d measurable on K . Hence ~ is integrable over K according to L e m m a 1.5.

2. Case: 0 = s < t . Then measurable and locally bounded function ~D exist on C v such t h a t

3 e~(t.v)

Then ~0#o/is a measurable a n d bounded function on K and 1"@) = ~ (~Bol) l*(~p).

3e~:(t,p)

According to L e m m a 1.6, the differential / 1 \ ~ 1

is integrable o v e r ' K . Hence ~ = ~#~(t.v) ~ is integrable over K.

3. Case: 0 =t < s . B y conjugation, this follows from case 2.

9 - 662905 Acta mathematica. 118. Xmprim6 le 12 avril 1967.

132 Wr~,~ELM STOLL

4. Case: O<s and O<t: Measurable and locally bounded function ~0aB exist on C ~ such t h a t

~ = 5 Y. ~ ^ $ ~ .

r #e~.(t,~)

Then /*(~0) -~ ~ ~. (~0~o/) ]*(~ A ~p),

where each ~0~pof is bounded and measurable on K. According to Lemma 1.4 is

= (log

1]" 1__1__

1/11 [/I s+'

(~ot)./*(~o~o)oz

integrable over K. Hence ~ = ~,E~(s.~) ~pE~(t.~)~p is integrable over K.

Return to the original assumptions t h a t K is a compact subset of a complex mani- fold M. Define

a(~) = {zlzEK with [/(z)[ ~<Q}.

If 0 < Q < 1, then

S(Q) = [ 121

J A (Q)

exists. Because A(p')~_A(p) if O<e' < 0 < 1 and because No<o<1 A(p)~/-1(0)N K is a set of measure zero. S(~)~O for 0 ~ 0 . Because

Iz(e)l < f~,o Ihol lift < Bf~,o)121

= ^s(e) also I(Q)-~0 for ~-~0; q.e.d.

Now, a type of residue formula shall be proved, which will be very helpful later on.

Let V be a complex vector space of dimension m with an Hermitian product (]). Then the projective forms eo and eo~ are defined on V - { 0 } . On P(V) the forms ~ and ~ are defined. Let ~: V-{0}-+P(V) be the residual map. Then o)~=O*(~). Because degree

~ = 2p, for p ~> m is ~ = 0. Therefore,

o~ o = 0 on V - { 0 } i f p > m . On V - {0} is

i 1

d" log ]$] = i ( a - ~ ) 8 9 log ($[$)= ~ [ ~ ((d$[$)- ($[d$)).

Moreover,

9

= ( m - ~ ) ~ Isl ~ - ~ k~/ i , - - 2 ) ~ Isl ~

A (dsl$)n ($1d$)

A G E N E R A L F I R S T M A I N T H E O R E M O F V A L U E D I S T R I B U T I O N . I 133 Hence d z log [3[ A a)m-t=[~[2-2md~ log [~[ A Vm-1.

Lv.M~A 1.8. Let M be a complex mani/old o~ pure dimension m. L e t / : M - ~ O ~ be a holomorphic, q /ibering map with q = m - p and O<p<~m. Let H be an open subset o / M with compact closure H. Let g be a continuous di//erential /orm o/ bidegree (q, q) on M. Let

T be the support o / g in I ~ - H . Suppose that/-1(0) f) T is a set o/measure zero on/-1(0) i / q > 0 and that/-1(0) f~ T = O i/ q = 0 . Suppose that a constant B > 0 a n d / o r every ~ i~.

0 <~ < 1 a/unction go o/class C ~ on R are given such that 1. For x E R and 0 < Q < I ks [go(x)[ <1.

2. For x E a and 0 < ~ < 1 ks

Ixg;(~)l

< B . 3. _Por x <~ 1~ ks go(x) = O.

4. For x > ~ ks gQ(x) = I .

~or m e c , and 0 <e < 1,

de/ine

~0(m) =gQ(] ~l)"

D4ins

I(e) = f d " log ]1] A d(~eo/) A/*(go-,) h Z.

2~P f ~

Then I(Q)-+(p_I)~T. nr_l(o)VrZ /or 9 ~ 0 ,

where, in the case q = O, X is a/unction and the integral means a sum:

f . v~z 7. ~(z; 0)

z(z).

f~f--l(0) ^{z e H}

Proo/. If q > 0, then define J by

J(lv) = ~ vfZ.

According to I I Theorem 3.9, J is continuous at 0EC p. If q = 0 , ~hen define J b y J(ro) = ~ vI(z; Yo) X(z).

z E H

According to I Proposition 3.2, J is continuous at 0 E C v. For 0 <~ < 1 is (1)

i(e) = foJ(w)

d" log ]m] h d)te(lv ) h (D~_I(m).

Then d2e=g~(]yO] ) (dF0[Y0) + (F0[d~o)

(1) For q > 0, seo II, Proposi$ion 2.9. For q = 0, see II, Proposition 2.8.

134 "WI~BW.LM STOLL

9 vp-1(W~

Hence d" log [iv[ A d~(ro) A co,_~ ( m ) - d "L log Iml A d~(ro) ;\ l ~ i 1

4 I~l ~.+x g~(liv[) A ( ( d W [ ~ ) - (~ldm)) A ((div[~ + (mld~o)) A V._l(~O)

= ~ go(Iwl) ~ 1 , i (aivliv)//(ivlaiv) A ~,,,_~(m).

Define Io(0) = f c ~ ( J ( m ) - JC0))d* log Ira] A d~(iv) A cOp_l(iv),

= fcpd ~

log lm ] A dgQ(iv) h

I I ( O ) f.o~ - 1 ( ( . o ) .

T h e n I(0 ) ~/o(O) + J(O) I1(0) for 0 < O < 1. It is

Q ( J ( i v ) - J(0)) 1 i ( d ~ l ~ ) A (~l~[d~l~) A ~)p_l(~0)

,<lwl<l (g(qiv)- g(0)) (g~ (liv] q)]ivl q) v~-l.

Define D = f89 <l~l<,]m~ ~+1 2 A (dmliv)A (ivldm)A v,_,Cm). 1 i

Then 0 < D < ~ . For every e > 0, n number

qo(e)

exists in 0 <0o(e) < 1 such that I iV I <0o(e) implies I J ( i v ) - J ( 0 ) ] <e. Therefore IJ(qiv)-J(0)[ < e if liv] <1 and 0<0<0o(~}. Hence

I Io(q) I <~ BDe

if 0 < 0

<Ode).

Hence

Io(0)-~0 for 0-~0.

f~ log d ~ oJp_l(iv )

:For I 1 is I I ( 0 ) = l~<[ro[<~ d"

l v I^

^A

= - ( d(Xqa" log [to[ A

r

JI

because ddXlog [ iv ] A OJp_l(iv ) = -- 2(D A (Dp_ 1 = -- 2pc% --- 0.

Let S o be the sphere of radius 0 with center at 0eC p oriented to the exterior of {$[ ]$[ <0}.

~tokes' Theorem implies

A G E N E R A L F I R S T M A I N T H E O R E M O F V A L U E D I S T R I B U T I O N . I 135

11(5) = - fsQd"

log [IV l/X cov_x(IV)

= - -

fsdTl~ ^IIVl

^{A ~_~(IV)}

= - fs d'

^log

I ml

A Vp-l(~).

i

If j :S 1"4 C p is the inclusion map, the j*(d x log 1/] IV ] A vv_~(IV)) is the euclidean volumo element of S 1. Hence

I~(5)=(p_l)!

if 0 < 5 < 1 .

3"~ ~

Hence 1(5) = lo(5)+J(0) 11(5)-* (p _ 1)! J(0) for Q-~0; q.e.d.

2. The intersection m, mher

Let V be a complex vector space of dimension n + l . Let P(V) be the associated projective space and let

Or: V - { 0 } ~ P ( V )

be the associated projection such t h a t

~l(sv(a)) = { z a l 0 # z e c } . Let V[p] be the p-folded exterior product of V. Define

& + l ( V ) =

{CoA

. , , /~

CpI {~/~E V) CZ

V[p + 1].

Then the Grassmann-manifold

r162 = 5 v,~+1~(&+~(V) - {0)) is a smooth, connected, compact submanifold of P ( V ~ + 1]),

If Co .... , cp are linearly independent vectors of V, then E ( c o . . . c~) -- {3]~ A CoA ... A C~ = o }

is the (p + 1)-dimensional complex subspace of V which is spanned by (Co ... c~). If ~ E q~(V) then E(~)=E(co ... cp) is well-defined by ;r This map E of (~(V) onto the set of all (p+l)-dimensional complex suhspaces of V is bijective. For 7Eq~'(V) define /0(F)=~(Co .... , c~)=sv(E(y)-{0}), where y=Sv[v+ll(CoA... A C~). Then E defines a bijective map of q~(V) onto the set of p-dimensional complex planes of V.

aL(V) ={

(Co ... c,) I Co A ... ~, c~ # 0 }

136 w t ~ . ~ STOLn

is the set of all bases of V. I t is the general linear group of V a n d is the complement of a thin analytic subset of V x ... • V (n-times).

Now, a local coordinate system of ( ~ ( V ) shall be introduced:

LV.MMA 2.1. Let a={ao ... a,)EGL(V) be a base o] Y. Let 0 ~ < p < n . Define

= Q v ~ +l](ao ^ . . . ^ o~) ~ ~ ( V ) . Define

Zo = {ml m e @ + l ( v ) with ro ^ ~,+, A... ^ u , , o ) . Z = ev~+l~(Zo).

Then Z is an open neighborhood o / a . A holomorphic map r o " - ~ " ~ + " ~ & + ~ ( v ) - {o}

is defined in the following way: Talce 36C(n-m(r+l) then 3~=(z~.~+x ... z~n)6C n-~ exists with 3 = ( $ o . . . . ,

$~).

De/ine c~ = ~ z,,a, /or p = 0 . . . p.

v f p + l

~ e t ~o(~) = (ao + Co) ^... ^ (a~ + c~).

T h e n ~ = ~V[p+l]O~o : C ("-D)(p+I) --)-Z

is a bihohm~rphlc map onto the open neighborhood Z o / ~ = ~(0) in ~/'(V).

Proof. 1. Denote 0 = 0 v~ +;]. Clearly, % + Co .... , ap + cn arc linearly independent. Hence

~o a n d ~ (into [~F(V)) are well-defined a n d holomorphic. Obviously, Z o is an open neigh- borhood of ~ A ... A ~ in (~+I(V). Hence Z is an open neighborhood of ~=~(0) in (~(V).

2. I f 3=($o .... , $~)EC (n-m(n+x) with 3~EC "-~, then

~0($) A (lp+ 1 A . . . A (I n = (QO -[- CO) A . . . A (O.~-'l- Cp) A (~+1 A . . . A (In = % A . . . A fin =~=0 Therefore, ~0 m a p s into Z o a n d ~ m a p s into Z.

3. The m a p ~ is surjective: T a k e flEZ. Then fl=~(b) with 13----l)0A...Al~p~=0 a n d 1~ ~ V and

~0 A ... A ~n A 0~+~ A ... A a n * 0 .

Now, the following s t a t e m e n t Sq shall be proved b y induction for q in 0 , . < q < p + l : Sq: "vectors C/,q = ~ cuq, a~ and a number

bq~

C ez~8$ such tha~

Y=q

b=bq(ao+Co, q) h

^{. . .} A ( 0 q _ 1 + Cq_l.a) A Ca.qA ... h Cp.q".

A GENERAL FIRST MAIN THEOREM OF VALUE DISTRIBUTION. I 137 I f q = 0 , choose C~o=b~ for p = 0 ... p a n d b o = l . T h e n S o is true. Suppose t h a t Sq is true.

T h e n Sq+ 1 shall be p r o v e d if q + 1 ~<p + 1. T h e n

0 ~ A O0+ z A ... A an = bqAao A ... A fin,

where A is c o m p u t e d t h a t way: L e t E~ be t h e u n i t m a t r i x with r rows a n d columns. L e t

~ s t be t h e zero m a t r i x with s rows a n d t columns. Define

T h e n

Cq_ I, p

Eq o, q

A = d e t ~ r - q + l , q C ~

c O ,

q-l,n ^{p + l x}

t~'q' P+I | = d e t qq

~q. n ~ Cpp.

/

H e n c e %,q,q:~O for some ~u in q<~#<~p. B y changing t h e e n u m e r a t i o n of cq,q ... c~,q, it can be a s s u m e d t h a t ca, q, q :pO. H e n c e ca, q = ca, q. q((~q*q- Cq,q+l) where

Cq, q+l ~ ~ Cq.q+l,t,O~*

~=q+l Define bq+ 1 = Ca.q. q bq. T h e n

= bq+l(flo q- to.q) A ... A ((lq_l-JrCq_l.q) A ((lq'q-Cq.q+l) A Cq+l, q A ... A Cpq.

F o r p =pq define

C.u.q+l=C.u.q--C.a.q.q(QqJf'Cq.q+l) -~" ~,, C.a.q+l.l,0-1,.

~=q+l

T h e n - - - - bq+l((lo q" Co. q+l) A A (O.q-~ Cq. q+l) A Cq+l. q+l A ... A Cq+ 1.p.

H e n c e Sq+ 1 is proved. Especially Sp.t. 1 is t r u e

~1 = bp+l(Q 0 .q- Co.p+i) A . . . A ((Lp --~ Cp.p.l_l)

w i t h C~o p+l ~ ~ C~p +lv av-

1,~p+l

Define ~=(%.r+1.r+1 ... %~+I..) for p = O ... p and ~=(3o ... 3r)EC(n-~)(~+l). Then b = b~+z~o(~) and ~(~)=e(~o(~))=~(b)=ft. Hence ~ is surjective.

4. T h e m a p ~ is injective: L e t

~ = (z~+l ... z~n) a n d e~ = (v#r+l ... v~n) for p = 0 ... p a n d

138 W I I , H E T , M S T O I ~

= (~o . . . ~ ) , v = (Vo ... v~) such t h a t ~($)=~(V). Define

C~,= ~ z~,a a n d

~ffip+l

for/~ = 0 ... p. A n u m b e r u =t=0 exists such t h a t

~ ' - p + l

= (ao +Co) A ... A ( ~ + c~) = U(ao + ~o) A ... A

(% +~).

T h e n ~ A ~+1 A ... A (In = ao A... A an = Uao A ... A a~ =t=0.

H e n c e ~ = 1. T a k e p with 0 ~ p ~ p . T h e n

(a~ + ~ ) A (% + Co) A ... ^ ( ~ + c~) = 0 . Because (% + Co) A... A (% + ~) 4 0 , n u m b e r s a~Q exist such t h a t

a ~ + ~;~ -- Q=~o a ~ (% + O . T h e n

0 ~:ao A ... A an = a0 A ... A a~-i A (a~ + ~ ) A a~+l A ... A an = a ~ a o A ... A an.

H e n c e a ~ = 1 if 0 ~ p ~ p . T a k e ~ ~:/z in 0 ~ ~ p . T h e n

0 = a~ A ao A ... A a~_~ A a~+~ A ... A an = ( % + ~ ) A ao A ... A ax-~ A a~+~ A ... A an ap~ a~ A % A ... A fl~-i A Q~+I A ... A On.

Hence a ~ = 0 if p ~:~. Therefore a~ + ~ = a~ + C~ or ~ = C~ which implies I~ = ~ f o r / ~ = 0 .... , p. H e n c e I~ =~. Therefore ~ is injective; q.e.d.

L e t M be a pure m-dimensional complex manifold. L e t F be a complex v e c t o r space of dimension n + l . Suppose t h a t 0 ~ p < n . L e t / : M ~ P ( V ) be a holomorphic m a p . T h e n

F~ = F ~ ( I ) = {(z, ~ ) l l ( z ) e & ~ ) , where (z, ~ ) ~ M •

is said to be the graph o / o r d e r p o / f . Obviously, F0(/) is t h e usual g r a p h o f / . Define z~,:F~(/)-+(~'(V) b y ~ f ( z , ~) =

~ : F ~ ( / ) ~ M b y ~ ( z , ~) = z . I f ~ e ~ ' ( V ) , t h e n

~i'(~) = {(~, ~)l z e M , 1(~) e & ~ ) } = l-'(&~)) • D } . H e n c e

(1) ~-~(~) = 1-~(~(~)) x

{~}

(~) ~ ( ~ / ' ( ~ ) ) = l-~(&~)).

A GENERAL FIRST MAIN THEOREM OF VALUE DISTRIBUTION. I 1 3 9

LEMMA 2.2. Let M be a pure m.dimensional complex mani/old. Let V be a complex vector space o] dimension n + l . Suppose that O<.p<n. Let ] : M ~ P ( V ) be a holomorphic map. Then the graph $'~(]) o/ order p o / / i s a smooth complex submani/old o / M • ~ ( V ) with pure dimension m + p ( n - p ) .

Remark: A local "coordinate system" at (a, o 0 E F~(]) can be obtained the/ollowing way:

1. Step. Let

e: V - {O}-+P(V) e: Vip + 1 ] - {O}-,-P(V[p + 1])

be the natural projections. Pick any base (ao ... a,) E GL(V) such that Q(ao A ... A %)= o~ and ](a)

=•(ao).

2. Step. Define

Vo = 00 + E ( f l l . . . fin) = {~o + ~.o.1~,, e e } p=l

Then fro=~(Vo) is an open neighborhood o/ /(a) in P(V). The map ~o=~[ Vo: VolVo is biholomorphic.

3. Step. Pick any open neighborhood A o / a such that/(A)~_ fr o.

4. Step. Define a holomorphic map

~:A • r

by the ]oUowing construction: Take (z, 3) e A • C " ( " - ~ then 3 = (31 . . . 3~) with 3~ = (z~.~+l . . . % . . ) e C "-~.

n

Define Sol(/(z)) = fio +u~=o/~(z) fiu,

C~=Cu(3) = ~ z~a, for / z = l ... p,

v = p + l

co = Co(Z, 3) = ~ /.(~) c ~ - ~ / , , ( ~ ) c.,

v=~v+l ,a=l

~(z, 3) = o(fio + co) A ... A (o~ + c~)), 5. Step. For (z, 3) EA • C ~("-m, define

a(z, 3) = (z, ~(z, 3 ) ) ~ M x ( ~ ( V ) .

Then B = a ( A • (n-m) is an open neighborhood o] (a, a)=a(a, O) in F~(]). Moreover, a :A x C ~(~-m ~ B is biholomorphic. Moreover, i] Z is the neighborhood o / ~ which was intro- duced in Lemma 2.1, then

B = (A • z ) o PA/).

1 4 0 W ~ L H E L M STOLL

Proo].

Clearly, if t h e s t a t e m e n t s of t h e R e m a r k are p r o v e d , t h e L e m m a 2.2 is also proved. T a k e (a, ~) E F~, t h e n t h e R e m a r k shall be p r o v e d . Now, % E V - {0} w i t h ~(%) = / ( a ) exists where / ( a ) E E ( x ) a n d ~ ( ~ ) = O ( E ( ~ ) - { 0 } ) . H e n c e % E E ( x ) . Therefore, a b a s e a = (% ... an) E

GL(V)

c a n be p i c k e d such t h a t ~(% A... A ~ ) = ~. This completes S t e p 1. S t e p 2, S t e p 3 a n d S t e p 4 are trivial a n d ~ is holomorphic.

F o r S t e p 5, define B = (A x Z) A ~ ( ] ) , where Z is t h e n e i g h b o r h o o d of ~ which w a s defined in L e m m a 2.1. T h e n B is a n open n e i g h b o r h o o d of (a, ~) in ~=(/). H o l o m o r p h i c functions ]~ exist on A such t h a t

e~(/(z))

⁼

ao+ ~

It(z) ~,,,

tu=l

H e r e is eo ~(](a)) = eo ~ (~t) + %. H e n c e

]#(a)

= 0 for #t = 1 ... ~t. Now, define/~: A x c2(n-m C ~v+i)~-v~ b y t h e following procedure: T a k e

(z,

3)EA x C2 ~-v). T h e n

3 = ($1 ... 3p) w i t h 3g = (zg~+l ... zg-) EC"-v"

Define 3o = 3o( z, 3 ) = (Zo.r+l ... Zo.,) b y

zo, = ~ ( z , 3) = l , ( z ) -

~ Ira(z) z.,.

I.t-1

Set/5(z, 3 ) = (z, 30, -.., 3v)EA • C (l~+l){a-l~}. Obviously/~ is a holomorphic, injective m a p w i t h a n J a c o b i a n of r a n k m + p ( n - p ) . Hence/~(A • C2 ("-v)) = B ' is a s m o o t h c o m p l e x s u b m a n i - fold of A x {3 (v+l)(n-m, w i t h dim

.B'--m +p(n-p).

Now, ~: C cp+i)("-~)-~Z is biholomorphie. H e n c e

= Id~ x ~ : A x r --*A x Z

is a biholomorphie m a p o n t o a n o p e n s u b s e t of M x ~ ( V ) . H e n c e ~ o ~ : A x

c~(n-m-~'A x Z

is a holomorphie, injective m a p of r a n k m + p ( n - p ) a n d its i m a g e ~ ( B ' ) = B " is a s m o o t h c o m p l e x s u b m a n i f o l d of A x Z w i t h p u r e dimension m + p ( n - 1 0 ) . Now, ~o~ = ~ a n d B ~ = B is claimed:

F o r (z, 3) E A x C ~(n-~) is 3 -- (31 . . . 3=) w i t h 3~ = (z~. v+l . . . zv,). Define

n

C~= =~+xz~,o, for # = 0 , 1 . . . p.

T h e n 3o = (Zo. n+l . . . zo. ,) w i t h zo, =- h(z) - ~.~-i fz(z) z~,. H e n c e

CO ~-~'~'= ~u*l ~'=p +1 I,=p+l

A G E N E R A L F I R S T MA'I~T T H E O R E M O F V A L U E D I S T R I B U T I O N . I 141 Therefore a(z, $) = (z, ~(z, $)) = (z, O((Oo + Co) A ... A (% +c~))

=

~(~,

^{3 o . . . . ,}

$~)

⁼

~(fl(~, $)).

Hence ~--~~

U s i n g the s a m e notation, it is

e#~(l(z))

^A (ao+ Co) A ... ^ (a~+ q,)

~=I v = p + l / ~ 1

= ~ ]r (a. --]- c/t) ^ ( 0 0 -~" CO) A (C[ 1-4- Cl) A . , , A (~I~ "Jf- Cp) = O.

H e n c e / ( z ) E ~(~(z, $)), w h i c h implies

~(z, $) = (z, ~(z, $))e F~.

H e n c e , a = ~ ' o f l is a n injective, h o l o m o r p h i c m a p into B; hence B'___ B.

T a k e a n y (z, ,/) e B. T h e n $ = ($o . . . 3v) e t3 cv+l~c~-p~ w i t h Sg = (z~r+x . . . zg,,) exists such t h a t r =r/. D e f i n e c~ = ~2=v+~ z ~ a ~ for g = 0 , ..., io. T h e n

0((ao + Co) ^... ^ ( ~ + c~)) =,~.

H e n c e offi(/(z)) ~ E(,/) -- E(ao + Co ... % + cv).

n

Hence ao + ~J.(~)a._ --~og.(a._ + c.),

which implies go = I and g~ =f~(z) for/~ = I ... p and

/ ~ = P + l /~=0

n p p

Hence Zo~ = ] , ( z ) - - ~ l t , ( z ) z m . bt~o

Therefore fl(z, 31 . . . Sv) = (z, $o . . . Sv) -- (z,

$)

a n d a(z, $i . . . St) = ~(fl(z, $1 . . . Sv) = (z, ~($)) = (z, 7).

Therefore, a is surjeetive. H e n c e a : A • is a b i h o l o m o r p h i c map. N o w , ],~(a) = 0 f o r / s = 1 ... /9, implies

a(a, o) = (a, ~(a, 0)) = (a, e(ao ^ ... ^ a~)) = (a, ~),

q.e.d.